Oscillation and Asymptotic Behaviour of a Higher-Order Nonlinear Neutral-Type Functional Differential Equation with Oscillating Coefficients

Abstract

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type: ${[y\left(t\right)+p\left(t\right)f\left(y\right(\tau \left(t\right)\left)\right)]}^{\left(n\right)}+q\left(t\right)h\left(y\right(\sigma \left(t\right)\left)\right)=0$, $t\ge t_0$, $t\in ℝ$, where $p\in C\left(\right[t_0,\infty ),ℝ)$, ${\text{lim}}_{t\to \infty }p\left(t\right)=0$, $q\in C\left(\right[t_0,\infty ),{ℝ}^{+})$, $\tau \left(t\right)$, $\sigma \left(t\right)\in C\left(\right[t_0,\infty ),ℝ)$, $\tau \left(t\right)$, $\sigma \left(t\right)<t$, ${\lim }_{t\to \infty }\tau \left(t\right)$, $\sigma \left(t\right)=\infty$, and $f,h\in C(ℝ,ℝ)$. We obtain sufficient conditions for the oscillation of all solutions of this equation.